Question Number 124398 by bounhome last updated on 03/Dec/20
$$\int\frac{\mathrm{2}{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}=? \\ $$
Answered by Dwaipayan Shikari last updated on 03/Dec/20
$$\int\frac{\mathrm{2}{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}\:\:\:\:\:\:\:{x}^{\mathrm{2}} ={t}\Rightarrow\mathrm{2}{x}=\frac{{dt}}{{dx}} \\ $$$$=\int\frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}={log}\left({t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}\right)={log}\left({x}^{\mathrm{2}} +\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\right)+{C} \\ $$